Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (3): 949-962.doi: 10.1016/S0252-9602(10)60092-6

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VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

 WEN Huan-Yao, DING Shi-Jin   

  1. School of Mathematical Sciences, |South China Normal University, Guangzhou 510631, China
  • Received:2007-10-19 Online:2010-05-20 Published:2010-05-20
  • Supported by:

    The project is supported by the National Natural Science Foundation of China (10471050), the National 973 Project of China
    (2006CB805902), University Special Research Fund for Ph.D Program (20060574002), and  Guangdong Provincial Natural Science Foundation (7005795,  031495)

Abstract:

In this article, using coordinate transformation and Gronwall inequality, we study the vortex motion law of the anisotropic Ginzburg-Landau equation in a smooth bounded domain $\Omega\subset{\bf R}^2$, that is, $ {\partial_tu_\varepsilon=\sum\limits_{j,k=1}^2(a_{jk}\partial_{x_k}u_\varepsilon)_{x_j}+\frac{b(x)(1-| u_\varepsilon| ^2)u_\varepsilon}{\varepsilon^2},x\in\Omega}$, and conclude that each vortex $ {b_j(t)~(j=1,2,\cdots ,N)}$ satisfies $ \frac{{\rm d}b_j(t)}{{\rm d}t}=-\big(\frac{a_{1k}(b_j(t))\partial_{x_k}a(b_j(t))}{a(b_j(t))},$ $ \frac{a_{2k}(b_j(t))\partial_{x_k}a(b_j(t))}{a(b_j(t))}\big),$ where $ {a(x)=\sqrt{a_{11}a_{22}-a_{12}^2}}$. We prove that all the vortices are pinned together to the critical points of $a(x)$. Furthermore, we prove that these critical points can not be the maximum points.

Key words: Anisotropic Ginzburg-Landau equation, Gronwall inequality, vortex dynamics

CLC Number: 

  • 35A21
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